Piecewise Function Research Articles

Research on time-varying path optimization for multi-vehicle type fresh food logistics distribution considering energy consumption

With the increasing demand for fresh food markets, refrigerated transportation has become an essential component of logistics operations. Currently, fresh food transportation frequently faces issues of high energy consumption and high costs, which are inconsistent with the development needs of the modern logistics industry. This paper addresses the optimization problem of multi-vehicle type fresh food distribution under time-varying conditions. It comprehensively considers the changes in road congestion at different times and the quality degradation characteristics of fresh goods during distribution. The objectives include transportation cost, dual carbon cost, and damage cost, subject to constraints such as delivery time windows and vehicle capacity. A piecewise function is used to depict vehicle speeds, proposing a dynamic urban fresh food logistics vehicle routing optimization method. Given the NP-hard nature of the problem, a hybrid Tabu Search (TS) and Genetic Algorithm (GA) approach is designed to compute an optimal solution. Comparison with TS and GA algorithm results shows that the TS-GA algorithm provides the best optimization efficiency and effectiveness for solving large-scale distribution problems. The results indicate that using the TS-GA algorithm to optimize a distribution network with one distribution center and 30 delivery points resulted in a total cost of CNY 12,934.02 and a convergence time of 16.3 s. For problems involving multiple vehicle types and multiple delivery points, the TS-GA algorithm reduces the overall cost by 2.94–7.68% compared to traditional genetic algorithms, demonstrating superior performance in addressing multi-vehicle, multi-point delivery challenges.

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

Modified Shooting with Discretization Validation for Non-Standard Optimal Control Problem: Case Study for Four-Stage Royalty Payment Problem

A modified shooting method augmented by discretization validation is presented in this paper to address the challenges inherent in non-standard optimal control problems. Specifically, a specific case study involving four-stage royalty payment functions is focused on, with the aim of effectively optimizing these complex, non-differentiable functions. The shooting method is adapted and enhanced to compute optimal solutions, and its accuracy is rigorously confirmed through discretization techniques. The primary objectives involve maximizing the performance index and determining optimal control strategies for scenarios where the final state variable remains unknown. In this study, the royalty payment is defined as a four-stage piecewise function. The incorporation of piecewise royalty functions introduces non-differentiability at specific time intervals, necessitating innovative approaches for finding optimal solutions. This, in turn, leads to the utilization of the continuous hyperbolic tangent (tanh) function to address the non-differentiability issue. A hybrid shooting method, combining the Newton and Golden Section Search methods, is employed in the C++ programming language to compute the unknown final state value. A new natural boundary condition, based on established theory, is introduced to further facilitate the investigation. Discretization methods such as Euler, Runge-Kutta, Trapezoidal, and Hermite-Simpson approximations are employed for validation. The validation process entails the use of the AMPL programming language with the MINOS solver. Comparative analyses reveal that the modified shooting method yields more accurate optimal results than discretization methods, thus demonstrating the method’s effectiveness in addressing non-standard optimal control problems. The significance of fundamental theories in addressing real-world challenges is underscored by this research, providing valuable insights for future researchers exploring mathematical approaches in similar contexts. The study contributes to the continued relevance of the academic field, particularly in science and mathematics education.

A high precision 3-D tropospheric delay model over China using ERA5 data

The zenith tropospheric delay (ZTD) is an important error in Global Navigation Satellite System (GNSS) navigation and positioning. The current most empirical tropospheric delay models are two-dimensional with low spatial and temporal resolution and cannot capture high-frequent accurate ZTD variations in small areas and short periods, particularly in Mainland China with diverse climate changes and large terrain differences. In this paper, a high precision three-dimensional (3-D) tropospheric delay grid model (ZTD_3D) over China with a horizontal resolution of 0.25° × 0.25° and a time resolution of 1 hour was established with the piecewise function describing the height change of ZTD from 2016-2019 fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis data (ERA5). The performance of the proposed model is verified by using radiosonde data and ZTD products from the Crustal Movement Observation Network of China (CMONOC), as well as the global pressure and temperature models (GPT2, GPT3-1 and GPT3-5). Results show that the RMS of ZTD_3D model is 2.04 cm when compared to radiosonde observations, and the accuracy is 55.1 %, 54.7 % and 63.1 % higher than that of GPT3-1 model, GPT3-5 model and GPT2 model, respectively, while the RMS of ZTD_3D model is 3.66 cm when compared to GPS ZTD from CMONOC, and the accuracy is 1.6 % and 29.7 % higher than that of GPT3-5 mode and GPT2 model, respectively. The application of the ZTD_3D model in GNSS precise point positioning (PPP) has showed accuracy improvement in the vertical direction. The model proposed in this study can provide accurate ZTD information and a reliable tropospheric delay correction model for precise GNSS positioning.

The analytical curvature distribution model of columns and mathematical solution for pushover analysis

AbstractThe acquisition of a flexural backbone curve for columns primarily relies on finite element analysis and experiments, of which the complexity and high cost are significant challenges. Hence, this study proposes an analytical curvature distribution model (CDM) along the full height of the column, which is the key point of formula derivation and theoretical solution of the column backbone curve. The proposed CDM is a piecewise function model composed of linear and quadratic functions, with the characteristics of continuity and differentiability at the intersection point. The deformation compatibility equations, equilibrium equations, and material constitutive equations based on the variables of CDM are constructed to form an equation system, the solution of which can be theoretically determined by an iterative algorithm. Furthermore, 155 test specimen columns of different materials, for example, reinforced concrete, high‐strength reinforced concrete, and shape memory alloy are used to validate the proposed CDM and mathematical solutions of pushover analysis through three crucial indicators, that is, goodness of fit of backbone curve, ultimate displacement, and peak force, indicating strong practicability, high accuracy, and wider applicability. This theoretical method can be applied to deal with the key issues of interest during pushover analysis, that is, predicting the flexural backbone curves with different materials, determining the curvature distribution throughout the entire process of pushover analysis, and characterizing the evolution process of the plastic region.

Estimating soil water retention curves over the entire saturation range: A thermal conductivity-based approach

The relationship between soil water content (θ) and suction (h, referring to the absolute value of pressure head), is described by the soil water retention curve (SWRC). Our earlier research (Fu et al. (2021, 2023a) [J. Hydrol.127171]; [J. Hydrol.129898]) recognized underlying correlations between SWRCs and soil thermal conductivity (λ) versus θ curves, and developed methodologies to ascertain the parameters of the van Genuchten (vG) equation using λ(θ) measurements, described by the Ghanbarian & Daigle (GD) equation, and basic soil characteristics. Limitations intrinsic to the van Genuchten equation restrict the GD-vG approach to generate precise estimates only in the wet and medium suction range, specifically h ranging from 0 to 150 mH2O. The validity of these approaches in the dry region remains uncertain. In this study, we associated the Peters-Durner-Iden (PDI) model parameters to those of the GD model. An initial examination was performed on the linearization processes needed to derive the hydraulic continuity water content (θhc) from the capillary water component as characterized by the PDI model and to choose the suction at oven dryness (h0) based on PDI model performance. Subsequently, two piecewise functions and two pedo-transfer functions were formulated to compute the PDI model parameters utilizing soil porosity, particle size distribution, and GD parameters, based on a calibration dataset comprising 25 different soils. The new GD-PDI approach was subsequently assessed with six independent soils and juxtaposed with the previous GD-vG approach. The GD-PDI approach outperformed the GD-vG approach, particularly within the dry range.

A bidirectional pedestrian macroscopic speed model construction based on pedestrian microscopic simulation experiments.

Studies of macroscopic speed modeling of bidirectional pedestrian cross-flows have relied heavily on scenario experiments, but the data itself may be deficient because large-scale scenario experiments are not easy to organize and subjects may not be walking under normal conditions. In order to explore the possibility of using microscopic pedestrian flow simulations for macroscopic speed modeling of pedestrian flows, a series of two-way pedestrian cross-flow simulation experiments were designed. Bidirectional pedestrian flows are defined as Peds1 and Peds2. The crossing angle and pedestrian flow rate are used as variables, and a bidirectional pedestrian flows simulation is designed as an orthogonal experiment. The crossing angles range from 15 to 165 degrees, and bidirectional pedestrian flow rate range from 1 ped/s to 8 ped/s. A series of simulations are built and performed on the GIS agent-based modeling architecture (GAMA) platform. By analyzing the flow data of bidirectional flows in the crossing area, it is found that when the Peds1 density falls below a threshold, Peds1 speed is determined by pedestrians themselves and mainly remains in a free flow state; otherwise, the Peds1 speed decreases with density. The clear effects such as Peds2 density on the Peds1 speed cannot be determined. A piecewise function combined with a linear function and an exponential function is constructed as the Peds1 speed model considering the influence of the crossing angle. The calibration results show that the piecewise function should be better than the non-piecewise function. Compared to the results of established studies, the results in this paper have some differences. Therefore, the simulation method cannot completely replace the scene experiments. However, this approach can provide suggestions for subsequent refinement of the experimental program, as well as a feasible direction for the construction of a speed relationship for bidirectional pedestrian flows.

Experimental identification of the bending stiffness and damping of plates using the frequency-adapted virtual fields Method

Inverse vibroacoustic methods can be used to identify the complex bending stiffness of a plate from its vibratory response. This work focuses on the Virtual Fields Method (VFM) and the Force Analysis Technique (FAT). The VFM uses functions called virtual fields to solve the Principle of Virtual Work, a weak form of equilibrium, and identifies complex bending stiffness. Here, the virtual fields are defined as piecewise functions over a surface smaller than the plate (virtual window). FAT uses a finite-difference scheme to discretize the fourth-order spatial derivatives of the displacement in the local equilibrium of the plate and thus estimates the bending stiffness. The application of a finite-difference scheme creates a bias in the identified stiffness, which increases with frequency. The Corrected Force Analysis Technique (CFAT) corrects this bias in the high-frequency domain. In this study, it is proposed to apply the CFAT principles to the VFM to adapt the method, using the virtual window size, so it can be applied it in the high-frequency domain. FAT, CFAT, the VFM and the Frequency-Adapted VFM will be presented. The method was tested on experimental data to identify the complex bending stiffness of an aluminium plate partially covered with a damping material.

Convolutional neural network approach for fault detection and characterization in medium voltage distribution networks

Power outages significantly impact the power industry by disrupting social welfare and economic stability. Still, existing methods for fault detection face challenges due to load and network topology, conditions, and installed equipment. However, recent advances in artificial intelligence (AI) are enabling researchers to create alternative approaches for fault detection and location strategies. Therefore, this paper introduces a novel method for detecting, classifying, and locating faults in power systems through voltage waveform analysis using a convolutional neural network (CNN) integrated with the Piecewise Function Put Together (PFPT) algorithm for fault detection and fault zone localization in a power distribution network. Utilizing Park's transformation, noise reduction PFPT sine fitting, and CNNs, the proposed method distinguishes between 'healthy' and 'faulty' conditions. Simulation results reveal that while the voltage Park's vector time behavior of a healthy system remains stable, it exhibits circular or mixed patterns under faulty conditions. These patterns enable the identification of four types of short circuit faults—single-line-to-ground (LG), line-to-line (LL), line-to-line-to-ground (LLG), and three-line (3L) faults—by analyzing 3D voltage Park's waveforms at network buses. The study validates fault type identification through the observation of rotating Park vectors from sine fitting of time-based voltage waveforms. By converting 3D voltage waveforms into high-resolution images, the method utilizes a CNN for fault recognition, achieving an accuracy of 93.1%. This innovative approach underscores the robustness and precision of combining traditional electrical engineering techniques with modern AI.

Simulation studies of charging-up effect for single GEMs

Many experiments revealed a charging-up effect inside the Gas Electron Multipliers (GEMs) holes, resulting in variation of GEM effective gain. Charge accumulation on the insulator surface inside the GEM hole (charge process) is involved in the present simulation models. With only the charge process in the simulation, a longer and larger gain stability of GEM was observed while comparing the charging-up curves to references, even distinct in the relatively high conductivity of GEM insulator. This observation indicates that charge transportation within the GEM insulator (discharge process) is non-negligible; the charging-up stability depends on the dynamic balance of the charge and discharge process. Therefore, a discharge model is induced and expressed by a piecewise function based on the presence of a simulation framework. Charging-up curves are compared for single GEMs with two different structures under different gain values and input flux; the trend from simulation with the discharge process is consistent with those from references. Although the deviation of simulated charging-up curves is observed at the region started to be stable, indicating improvements in the discharge formulation, it can still predict the trend of stability and values of GEM effective gain. This study improves the present simulation model of the charging-up effect for single GEMs and provides a reference for other charging-up models.

A comprehensive bearing prognosis framework based on piecewise function stacking convolution auto-encoder and XGBoost algorithm

Abstract Accurately forecasting the remaining useful life (RUL) stands as a pivotal and formidable task within the realm of prognostics and health management. However, there is limited research that considers integrating early fault diagnosis during the bearing’s lifecycle with the prediction of its RUL. In this article, a comprehensive bearing prognosis framework based on piecewise function stacking convolution auto-encoder (AE) and XGBoost algorithm is proposed. To achieve this, an unsupervised piecewise function-based deep stacked convolutional AE was designed to construct the health indicator (HI) of the bearing for reducing the dependency on prior knowledge and furnishing a dynamic foundation for predicting RUL. The 4σ criterion based on HI’s increment was proposed for determining the fault occurrence time (FOT) of the bearing’s operational process. Subsequently, an XGBoost algorithm model was utilized to predict the RUL of faulty bearings. The efficacy of the bearing prognosis framework was validated by two real bearing test datasets. Results indicate the employed criterion of construct HI can flexibly adjust to various operational conditions and accurately pinpoint the bearing’s FOT. Furthermore, the findings demonstrate the proposed bearing prognosis framework achieves superior performance compared with several conventional anomaly detection methods.

A continuous-discontinuous coupling computational method for multi-material mixtures

Considering the important role of particle-reinforced composites (PRCs), and to accurately assess the mechanical properties of PRC with arbitrarily complex internal structures, this paper proposes a new continuous-discontinuous coupled computational method, i.e., the multi-material integrated analysis (MIA) method, by combining the advantages of the discontinuous deformation analysis (DDA) method, the numerical manifold method (NMM), and the meshless method, for the continuous-discontinuous dual state of this kind of materials when they are subjected to forces. The core idea of this method is by establishing a single physical cover (PC) on each particle and defining a unified and piecewise approximation function over it to achieve displacement coordination between the particles and the matrix. In this model, the particles can maintain the contact and separation characteristics like the blocks, while the matrix is able to achieve the bonding function by overlapping different covers. Two numerical integration schemes, namely single-point Gaussian integration and Monte Carlo integration, are proposed to address the integration issues in stiffness matrix calculations. Meanwhile by considering the similarity between NMM and meshless method interpolation function construction, this paper combines circular coverage and 0th-order moving least squares to construct the interpolation function to form the displacement approximation function for the whole solution space. Moreover, the method of lattice retrieval (also Nearest Neighbor Search, NNS) is invoked for particle contact determination to improve the efficiency of contact and coverage relationship determination. The validity of the model is verified by taking asphalt mixture, a typical PRC, as a demonstrative case study. The results show that the MIA method solves the one-sided problem that the existing algorithms usually only consider the continuous or discontinuous properties of PRC, and provides a new method for comprehensively analyzing the micromechanical response of PRC and solving its large deformation problem.

Modelling of Joule-Thomson cooling effect using a modified shift-DeepONet method for predicting hydrate onset during CO2 sequestration

Carbon Storage in underground depleted oil and gas reservoirs play a key role in reducing anthropogenic Carbon Dioxide (CO2) emissions around the world. Injection of CO2 at high pressure into a low pressure depleted gas formations containing methane (CH4) leads to Joule-Thompson (J-T) cooling, possibly resulting in injection blocking precipitation of CO2 or CH4 hydrates. Here, we present a deep neural operator-based approach to model the J-T effects during CO2 injection in depleted reservoirs to demonstrate the effectiveness of Deep Operator Networks (DeepONets) in modelling piece-wise functions with sharp changes in gradient for cases of varying permeabilities. We propose a modified version of the shift-DeepONet model using non-linear constructors in the neural network architecture to accurately model temperature and pressure profiles in space and time. The modified shift-DeepONet model achieves a higher accuracy in modelling temperature for permeability in the range of 1.01mD to 101mD with three to up to ten times improvement in reducing Relative L2 (RL2) norm and with a testing RL2 norm of 3.8 × 10−4 compared to 2.97 × 10−3 for the Vanilla DeepONet. Using the temperature and pressure profiles, estimates are made for hydrate onset in space and time. The proposed modified shift-DeepONet method achieves high accuracy in hydrate classification with precision values > 0.99, recall values > 0.86 and F1 scores >0.92 in classification of CO2 or CH4 hydrate onset during CO2 injection into depleted gas reservoirs.

Sampled-data control for Markovian switching neural networks with output quantization and packet dropouts

This paper explores sampled-data control for Markovian switching neural networks (MSNNs) with dynamic output quantization and packet dropouts. The primary goal is to construct a multi-mode, quantized sampled-data controller that ensures stochastic stability and H∞ disturbance-reduction performance of the closed-loop MSNN. A Bernoulli-distributed random variable with uncertain probability is introduced to characterize the incidence of packet dropouts. To describe potential mode inconsistencies that may occur between the MSNN and controller, an exponential hidden Markov model is employed. Furthermore, the quantizer’s dynamic scaling factor is intentionally built as a piecewise function to avoid the potential division-by-zero problem. A sufficient condition for stochastic stability and H∞ disturbance-reduction performance is proposed, utilizing a mode- and time-dependent Lyapunov-type functional and several stochastic analysis tools. Then, through decoupling nonlinearities, a numerically efficient approach for determining the desired controller gains and parameter range associated with the dynamic scaling factor is developed. In order to facilitate comparisons, the situation with no uncertainty in the probability of packet dropouts is studied, and both analysis and design approaches are offered. Finally, two simulation examples are provided to validate the effectiveness and applicability of the developed approaches.

Observational Constraints on the Aerosol Optical Depth-Surface PM2.5 Relationship during Alaskan Wildfire Seasons.

Wildfire is one of the main sources of PM2.5 (particulate matter with aerodynamic diameter < 2.5 μm) in the Alaskan summer. The complexity in wildfire smokes, as well as limited coverage of ground measurements, poses a big challenge to estimate surface PM2.5 during wildfire season in Alaska. Here we aim at proposing a quick and direct method to estimate surface PM2.5 over Alaska, especially in places exposed to strong wildfire events with limited measurements. We compare the AOD-surface PM2.5 conversion factor (η = PM2.5/AOD; AOD, aerosol optical depth) from the chemical transport model GEOS-Chem (ηGC) and from observations (ηobs). We show that ηGC is biased high compared to ηobs under smoky conditions, largely because GEOS-Chem assigns the majority of AOD (67%) within the planetary boundary layer (PBL) when AOD > 1, inconsistent with satellite retrievals from CALIOP. The overestimation in ηGC can be to some extent improved by increasing the injection height of wildfire emissions. We constructed a piecewise function for ηobs across different AOD ranges based on VIIRS-SNPP AOD and PurpleAir surface PM2.5 measurements over Alaska in the 2019 summer and then applied it on VIIRS AOD to derive daily surface PM2.5 over continental Alaska in the 2021 and 2022 summers. The derived satellite PM2.5 shows a good agreement with corrected PurpleAir PM2.5 in Alaska during the 2021 and 2022 summers, suggesting that aerosol vertical distribution likely represents the largest uncertainty in converting AOD to surface PM2.5 concentrations. This piecewise function, η'obs, shows the capability of providing an observation-based, quick and direct estimation of daily surface PM2.5 over the whole of Alaska during wildfires, without running a 3-D model in real time.

A method to standardize the temperature for near infrared spectra of the indigo pigment in non-dairy cream based on symbolic regression

Near infrared (NIR) spectroscopy is sensitive to physical conditions such as sample temperature, meaning that rapid detection methods based on NIR spectroscopy are significantly influenced by temperature. To address this challenge, symbolic regression was employed to mitigate the effects of temperature. The Weighted Windowed Adaptive Optimization algorithm was proposed and combined with the Sequential Projection Algorithm to extract temperature-related feature points and remove redundant data. Subsequent 3D modeling of these feature points revealed that absorbance alterations due to temperature comprised two distinct segments. Consequently, based on symbolic regression, the temperature standardization algorithm was devised to generate piecewise equations. This algorithm surpassed genetic programming and non-segmented methods in performance metrics. The piecewise function equations generated by the algorithm were used to regress the absorbance at different temperatures to the standard temperature. Non-dairy cream, with different indigo pigment contents, was temperature standardized using a piecewise function to obtain spectra at two standard temperatures; 18°C and 28°C. The r2 for the quantitative regression model improved from 0.71 to 0.95 at 18°C and from 0.63 to 0.85 at 28°C. The temperature standardization method offers interpretable equations for spectra that model the complex changes with temperature, factoring out the temperature variation, thereby facilitating the practical use of NIR spectroscopy in rapid detection applications.

N-level complex helical structure modeling method

This paper primarily explores the modeling method of n-level complex helical structures with coal mining machine cables as the research object. The paper first elaborately introduces the modeling method of n-level helix curves based on parametric equations and coordinate transformations, and compensates for the n-level helix curves with corrected pitch, which can obtain more accurate n-level helix curves and improve the accuracy of n-level helix curves modeling. Subsequently, based on this high-precision n-level helix curves modeling method, the paper elaborates on the method of solving pitch and twisting radius of multi-layer helical structure. Calculation scripts were written based on the above methods, which can be used to batch calculate the twisting radius and pitch of each layer structure in multi-layer structures when satisfying the conditions of in-layer tangency, inter-layer tangency, and extrusion deformation, and retain the actual results through logical judgment. Then, based on the above two methods, the paper developed a modeling method for braided structures based on piecewise functions containing fifth-order polynomials, which can effectively avoid the problem of insufficiently dense arrangement of braided lines and easy interference in traditional methods. Finally, a set of modeling tools was developed using C# and Python in Grasshopper to implement the modeling algorithm. Taking the MCPT-1.9/3.3 3120 + 170 + 4 * 10 coal mining machine cable as an example. The cable was modeled using both the method proposed in this paper and the traditional method. Comparative data shows that the method proposed in this paper can reduce errors by 3.31E6 times in the second-level and above helical structures. In addition this paper compares the standard line length, measured line length, and the line length established by the proposed model, showing that the relative errors are both less than 0.1941%. This paper provides a new, systematic, high-precision, and full-process cable modeling method, in which all parameters except the process parameters are accurately solved by equations. It lays a theoretical foundation for the high-precision simulation and intelligent sensing cables, which is of great significance for improving the safety, stability, and efficient development of the coal mining industry. The research results of the paper can not only be applied to the modeling of coal mining machine cables but also can be extended to the modeling of other complex multi-layer helical structures.

A fast and robust two‐point ray tracing method in layered vertical transversely isotropic media with strong anisotropy

AbstractA fast and robust two‐point ray tracing method was developed for layered vertical transversely isotropic media with strong anisotropy. Utilizing the Christoffel slowness equation, a novel generalized dimensionless ray parameter, , modified from the ray parameter (horizontal slowness), was proposed to efficiently and simultaneously determine the ray paths and travel times for direct and reflected quasi‐P, quasi‐SV and quasi‐SH waves. The Newton optimization algorithm was employed to solve the nonlinear offset equation accurately, resulting in rapid convergence to the true value. The inferred analytical equations show that the generalized ray parameter stabilizes the inversion process at large offsets. Additionally, a piecewise function was introduced to enhance the initial value estimation and calculation efficiency. The numerical results demonstrate that this novel approach can reduce the iteration error to 10−10 m in less than three iterations. Monte Carlo simulations further validated the effectiveness of the method for inferring the true ray paths at various offsets within complex velocity models. Furthermore, the method can address the triplication issue in quasi‐SV waves and exhibit robustness in strong‐layered vertical transversely isotropic media.

Adaptive fixed-time neural consensus control for a class of uncertain nonlinear multi-agent systems with full state constraints

This paper is concerned with the fixed-time consensus control problem for non-strict feedback multi-agent systems with asymmetric output constraints and full state constraints. Considering the feasibility of controlling execution, a novel practical virtual control signal is developed utilizing both saturation function and hyperbolic tangent function to ensure that this signal can remain within the same restricted range as the corresponding state variable throughout entire operation process. In backstepping steps, the design of ideal virtual control signal also adopts a different form of piecewise function than before, introducing high-order polynomial functions to avoid singularity problems in the derivation process. In addition, function approximation ability of radial basis function neural networks technique is applied to estimate uncertainties derived from the system functions and controller design procedure. Moreover, universal barrier Lyapunov function approach is improved for constructing an adaptive constrained synchronization control scheme. By fixed-time stability theory, it is shown that the tracking errors of the MAS converge to an adjustable region around the origin in a fixed time and the state variables always obey their constraints. And the upper bound of the settling time is merely dependent on design parameters, which is not affected by the initial states of MAS. The effectiveness of the proposed control strategy is shown by a numerical simulation example at last. Two scenarios are provided to demonstrate the advantages of the control protocol proposed in this paper.

Modeling of Granulation in Red Supergiants in the Magellanic Clouds with the Gaussian Process Regressions

The granulation of red supergiants (RSGs) in the Magellanic Clouds is systematically investigated by combining the latest RSG samples and light curves from the Optical Gravitational Lensing Experiment and the All-Sky Automated Survey for Supernovae. The present RSG samples are first examined for foreground stars and possible misidentified sources, and the light curves are sequentially checked to remove the outliers by white noise and photometric quality. The Gaussian process (GP) regression is used to model the granulation, and the Markov Chain Monte Carlo is applied to derive the granulation amplitude σ and the period of the undamped oscillator ρ, as well as the damping timescale τ. The dimensionless quality factor Q is then calculated through Q = π τ/ρ. RSGs around Q=1/2 are considered to have significant granulation signals and are used for further analysis. Combining granulation parameters with stellar parameters, robust scaling relations for the timescale ρ are established, while the scaling relations for amplitude σ are represented by a piecewise function, possibly related to the tendency of amplitudes in faint RSGs to converge toward a certain value. Comparing results between the Small Magellanic Cloud and the Large Magellanic Cloud confirms that amplitudes and timescales become larger with metallicity. In examining the scaling relations between the two galaxies, it is found that ρ is nearly independent of metallicity, whereas σ is more significantly affected by metallicity. The GP method is compared with the periodogram fitting of the granulations, and the advantages of either are discussed.